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We consider Hermitian random band matrices $H=(h_{xy})$ on the $d$-dimensional lattice $(mathbb Z/Lmathbb Z)^d$. The entries $h_{xy}$ are independent (up to Hermitian conditions) centered complex Gaussian random variables with variances $s_{xy}=mathbb E|h_{xy}|^2$. The variance matrix $S=(s_{xy})$ has a banded structure so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In dimensions $dge 8$, we prove that, as long as $Wge L^epsilon$ for a small constant $epsilon>0$, with high probability most bulk eigenvectors of $H$ are delocalized in the sense that their localization lengths are comparable to $L$. Denote by $G(z)=(H-z)^{-1}$ the Greens function of the band matrix. For ${mathrm Im}, zgg W^2/L^2$, we also prove a widely used criterion in physics for quantum diffusion of this model, namely, the leading term in the Fourier transform of $mathbb E|G_{xy}(z)|^2$ with respect to $x-y$ is of the form $({mathrm Im}, z + a(p))^{-1}$ for some $a(p)$ quadratic in $p$, where $p$ is the Fourier variable. Our method is based on an expansion of $T_{xy}=|m|^2 sum_{alpha}s_{xalpha}|G_{alpha y}|^2$ and it requires a self-energy renormalization up to error $W^{-K}$ for any large constant $K$ independent of $W$ and $L$. We expect that this method can be extended to non-Gaussian band matrices.
We consider Greens functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(mathbb Z/Lmathbb Z)^d$. The entries $h_{xy}=overline h_{yx}$ of $H$ are independent centered complex Gaussian random variables with
We consider $Ntimes N$ Hermitian random matrices $H$ consisting of blocks of size $Mgeq N^{6/7}$. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to blo
Consider $Ntimes N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W geq N^{3/4+varepsilon}$ for any $varepsilon>0$. In the bulk of the spectrum and in the large $N$ limit, we obtain the foll
We survey recent mathematical results about the spectrum of random band matrices. We start by exposing the Erd{H o}s-Schlein-Yau dynamic approach, its application to Wigner matrices, and extension to other mean-field models. We then introduce random
We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, $Wsim N$. All previous results concerning universality of no